Evolutionary Cooperation Dynamics with Mixed Strategy Updating of Combining Super-Rational Imitation and Aspiration Process

Understanding the mechanisms that generate and maintain cooperation remains a fundamental challenge in evolutionary game theory, with micro-level mechanisms playing a pivotal role in strategy evolution. Hofbauer’s theory suggests that imitation effects are more significant than genetic factors in the spread of successful strategies. Building on this framework, previous studies have explored various strategy update mechanisms, though they often focus on single mechanisms. This paper presents a novel mixed strategy update mechanism that integrates super-rational imitation and the aspiration process within a well-mixed finite population. Individuals select between these two strategies with a defined probability. We derived conditions that are conducive to the evolution of cooperation, and apply weak selection approximation to two specific cases: the prisoner’s dilemma and the stag hunt games. Theoretical analysis and numerical simulations reveal that cooperation is promoted when cooperators are super-rational, whereas defectors being super-rational impedes cooperation. Moreover, when cooperators exhibit low super-rationality, high aspiration levels support cooperation; conversely, when cooperators display high super-rationality, lower aspiration levels are more beneficial. This study provides important conditions for maintaining cooperation in various evolutionary settings, thereby extending the scope of evolutionary game theory.